A map is a two-dimensional representation of features seen on the substantially spherical surface of the earth. Unlike a terrestrial globe, which might be regarded as a spherical map, a two-dimensional map on a plane cannot depict the surface of the globe without distortion. Various "projections" have been developed in attempts to minimize the effects of these distortions or preserve some desirable aspect without distortion.
As used herein, the term "projection" refers to a systematic representation of intersecting coordinate lines on a flat surface upon which features from the curved surface of the earth or other celestial body are mapped in accordance with a prescribed mathematical relationship between the x and y coordinates of various points on the map and the latitudes and longitudes of corresponding points on the globe. The term "graticule" is used to refer to the systematic representation of coordinate lines in accordance with the said mathematical relationship, so that a "projection" is derived by plotting features on the "graticule."
The best known world map is that shown on Mercator's conformal projection, a navigation chart which, in effect, projects the spherical surface of a globe in a certain manner onto a surrounding cylinder of the same equatorial diameter, the cylinder then being unrolled to form a flat map. On such a map, the areal extent of features near the North and South Poles, such as Alaska, Greenland and Antarctica, is enormously enlarged relative to the areas in equatorial regions and relative to their true size on the earth.
Maps of this kind are not suitable for statistical purposes, such as depicting population densities. Therefore, a category of maps has been developed known as equal-area projections in which the areal extents of features on the map are the same in proportion as those of the corresponding features on the earth. Many such projections are depicted in the text "Equal-Area Projections for World Statistical Maps", by F. Webster McBryde and Paul D. Thomas, U.S. Government Printing Office, 1949, and the mathematical bases for the graticules of equal-area projections are also described and tabulated in that text. Known equal-area maps include "fusiform" types in which meridians converge to points at the top and bottom of the map, and "flat-polar" projections, which are a compromise with cylindrical projections, wherein the poles are depicted as straight lines, parallel to and shorter than the equator so that meridians converge towards the poles but do not meet at a point. In one type of such projection, known as Eckert's Projections, the poles are one-half the length of the equator. As described in the aforesaid text, other maps of this general category have been developed in which the poles are lines parallel to the equator, but shorter than in Eckert's Projections.
Implicit in equal-area projections are certain linear distortions of distances and distortions in shape. In other words, even though all areas are shown in correct size relative to the earth and to one another, some linear distances become distorted. As a result, the map length of, say, Greenland, may be increased relative to its width. This, in turn, may cause the shape of Greenland to be elongated or flattened, and different from its real or true shape, even though its areal extent is correct and it retains its proper relationship to that of other features on the map.
An attempt has been made to overcome those distortions with a composite heterolinear projection, i.e., by combining high latitude (northern and southern) portions of one projection with low and middle latitude (northern and southern) portions of another projection. The resultant composite projection, known as Goode's Homalosine, combines the sinusoidal equal-area projection for regions of latitudes 0-40 degrees north and south with the Mollweide homalographic equal-area projection for regions from 40-90 degrees latitude (north and south). However, even with this correction, there are significant distortions of the type described above. Also meridian curves are different in the two component projections and neither is flat-polar; the Sinusoidal projection is fusiform, with meridians as sine curves, and the homalographic projection is an oval, with eliptical meridians.